The old city is walkable — almost unavoidably so, since the lanes are too steep and narrow for anything except feet. The main sites (cathedral, Hanging Houses, museums) are all within a few minutes' walk of each other on the promontory. Allow for some serious uphill walking: the streets from the Plaza Mayor to the Casas Colgadas climb steeply, and the walk down to the gorge and back up is worthwhile exercise.
Between the new town (where the train and bus stations are) and the old city, the options are taxi (about €6), local bus, or a 20-minute uphill walk. The walk is manageable if you're not carrying luggage; the road winds up through successive hairpin bends.
Taxis are the most practical option for the AVE station (5 km from the old city) and cost around €8–10. There's usually a taxi rank on the Plaza Mayor.
City buses run between the new and old town, but the schedule is infrequent and not always reliable for visitors.
The new town (where most locals live) is flat and easy to navigate on foot. Cycling is possible on the valley roads but not practical in the old city given the gradients.